Properties

Label 2842.2.a.r.1.4
Level $2842$
Weight $2$
Character 2842.1
Self dual yes
Analytic conductor $22.693$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2842,2,Mod(1,2842)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2842, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2842.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2842.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.6934842544\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11348.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 406)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.723742\) of defining polynomial
Character \(\chi\) \(=\) 2842.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.76342 q^{3} +1.00000 q^{4} +2.47620 q^{5} +2.76342 q^{6} +1.00000 q^{8} +4.63646 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.76342 q^{3} +1.00000 q^{4} +2.47620 q^{5} +2.76342 q^{6} +1.00000 q^{8} +4.63646 q^{9} +2.47620 q^{10} +4.76342 q^{11} +2.76342 q^{12} -0.155665 q^{13} +6.84276 q^{15} +1.00000 q^{16} -5.23961 q^{17} +4.63646 q^{18} -1.44748 q^{19} +2.47620 q^{20} +4.76342 q^{22} +0.552516 q^{23} +2.76342 q^{24} +1.13155 q^{25} -0.155665 q^{26} +4.52223 q^{27} +1.00000 q^{29} +6.84276 q^{30} -7.45051 q^{31} +1.00000 q^{32} +13.1633 q^{33} -5.23961 q^{34} +4.63646 q^{36} -8.07935 q^{37} -1.44748 q^{38} -0.430166 q^{39} +2.47620 q^{40} -4.66518 q^{41} -3.63646 q^{43} +4.76342 q^{44} +11.4808 q^{45} +0.552516 q^{46} -4.55554 q^{47} +2.76342 q^{48} +1.13155 q^{50} -14.4792 q^{51} -0.155665 q^{52} -7.16329 q^{53} +4.52223 q^{54} +11.7952 q^{55} -4.00000 q^{57} +1.00000 q^{58} -6.05523 q^{59} +6.84276 q^{60} +9.12695 q^{61} -7.45051 q^{62} +1.00000 q^{64} -0.385456 q^{65} +13.1633 q^{66} +14.7906 q^{67} -5.23961 q^{68} +1.52683 q^{69} +13.6154 q^{71} +4.63646 q^{72} +0.439855 q^{73} -8.07935 q^{74} +3.12695 q^{75} -1.44748 q^{76} -0.430166 q^{78} -17.0107 q^{79} +2.47620 q^{80} -1.41260 q^{81} -4.66518 q^{82} -6.10346 q^{83} -12.9743 q^{85} -3.63646 q^{86} +2.76342 q^{87} +4.76342 q^{88} -10.1920 q^{89} +11.4808 q^{90} +0.552516 q^{92} -20.5889 q^{93} -4.55554 q^{94} -3.58426 q^{95} +2.76342 q^{96} +4.66518 q^{97} +22.0854 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - q^{3} + 4 q^{4} + q^{5} - q^{6} + 4 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - q^{3} + 4 q^{4} + q^{5} - q^{6} + 4 q^{8} + 9 q^{9} + q^{10} + 7 q^{11} - q^{12} + 7 q^{13} - 5 q^{15} + 4 q^{16} + 9 q^{18} - 2 q^{19} + q^{20} + 7 q^{22} + 6 q^{23} - q^{24} + 9 q^{25} + 7 q^{26} - 13 q^{27} + 4 q^{29} - 5 q^{30} + 7 q^{31} + 4 q^{32} + 19 q^{33} + 9 q^{36} - 12 q^{37} - 2 q^{38} - 15 q^{39} + q^{40} - 4 q^{41} - 5 q^{43} + 7 q^{44} + 44 q^{45} + 6 q^{46} + 11 q^{47} - q^{48} + 9 q^{50} - 16 q^{51} + 7 q^{52} + 5 q^{53} - 13 q^{54} - 3 q^{55} - 16 q^{57} + 4 q^{58} - 16 q^{59} - 5 q^{60} + 34 q^{61} + 7 q^{62} + 4 q^{64} - q^{65} + 19 q^{66} + 2 q^{67} - 18 q^{69} + 24 q^{71} + 9 q^{72} + 24 q^{73} - 12 q^{74} + 10 q^{75} - 2 q^{76} - 15 q^{78} - 9 q^{79} + q^{80} + 40 q^{81} - 4 q^{82} + 8 q^{83} - 24 q^{85} - 5 q^{86} - q^{87} + 7 q^{88} - 2 q^{89} + 44 q^{90} + 6 q^{92} - 55 q^{93} + 11 q^{94} + 20 q^{95} - q^{96} + 4 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.76342 1.59546 0.797729 0.603016i \(-0.206034\pi\)
0.797729 + 0.603016i \(0.206034\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.47620 1.10739 0.553695 0.832720i \(-0.313218\pi\)
0.553695 + 0.832720i \(0.313218\pi\)
\(6\) 2.76342 1.12816
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 4.63646 1.54549
\(10\) 2.47620 0.783042
\(11\) 4.76342 1.43622 0.718112 0.695928i \(-0.245007\pi\)
0.718112 + 0.695928i \(0.245007\pi\)
\(12\) 2.76342 0.797729
\(13\) −0.155665 −0.0431736 −0.0215868 0.999767i \(-0.506872\pi\)
−0.0215868 + 0.999767i \(0.506872\pi\)
\(14\) 0 0
\(15\) 6.84276 1.76679
\(16\) 1.00000 0.250000
\(17\) −5.23961 −1.27079 −0.635396 0.772186i \(-0.719163\pi\)
−0.635396 + 0.772186i \(0.719163\pi\)
\(18\) 4.63646 1.09282
\(19\) −1.44748 −0.332076 −0.166038 0.986119i \(-0.553097\pi\)
−0.166038 + 0.986119i \(0.553097\pi\)
\(20\) 2.47620 0.553695
\(21\) 0 0
\(22\) 4.76342 1.01556
\(23\) 0.552516 0.115207 0.0576037 0.998340i \(-0.481654\pi\)
0.0576037 + 0.998340i \(0.481654\pi\)
\(24\) 2.76342 0.564080
\(25\) 1.13155 0.226311
\(26\) −0.155665 −0.0305283
\(27\) 4.52223 0.870303
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 6.84276 1.24931
\(31\) −7.45051 −1.33815 −0.669076 0.743194i \(-0.733310\pi\)
−0.669076 + 0.743194i \(0.733310\pi\)
\(32\) 1.00000 0.176777
\(33\) 13.1633 2.29144
\(34\) −5.23961 −0.898586
\(35\) 0 0
\(36\) 4.63646 0.772744
\(37\) −8.07935 −1.32824 −0.664118 0.747628i \(-0.731193\pi\)
−0.664118 + 0.747628i \(0.731193\pi\)
\(38\) −1.44748 −0.234813
\(39\) −0.430166 −0.0688817
\(40\) 2.47620 0.391521
\(41\) −4.66518 −0.728578 −0.364289 0.931286i \(-0.618688\pi\)
−0.364289 + 0.931286i \(0.618688\pi\)
\(42\) 0 0
\(43\) −3.63646 −0.554556 −0.277278 0.960790i \(-0.589432\pi\)
−0.277278 + 0.960790i \(0.589432\pi\)
\(44\) 4.76342 0.718112
\(45\) 11.4808 1.71146
\(46\) 0.552516 0.0814640
\(47\) −4.55554 −0.664494 −0.332247 0.943192i \(-0.607807\pi\)
−0.332247 + 0.943192i \(0.607807\pi\)
\(48\) 2.76342 0.398865
\(49\) 0 0
\(50\) 1.13155 0.160026
\(51\) −14.4792 −2.02750
\(52\) −0.155665 −0.0215868
\(53\) −7.16329 −0.983954 −0.491977 0.870608i \(-0.663726\pi\)
−0.491977 + 0.870608i \(0.663726\pi\)
\(54\) 4.52223 0.615397
\(55\) 11.7952 1.59046
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 1.00000 0.131306
\(59\) −6.05523 −0.788324 −0.394162 0.919041i \(-0.628965\pi\)
−0.394162 + 0.919041i \(0.628965\pi\)
\(60\) 6.84276 0.883397
\(61\) 9.12695 1.16859 0.584293 0.811543i \(-0.301372\pi\)
0.584293 + 0.811543i \(0.301372\pi\)
\(62\) −7.45051 −0.946216
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.385456 −0.0478100
\(66\) 13.1633 1.62029
\(67\) 14.7906 1.80695 0.903477 0.428637i \(-0.141006\pi\)
0.903477 + 0.428637i \(0.141006\pi\)
\(68\) −5.23961 −0.635396
\(69\) 1.52683 0.183809
\(70\) 0 0
\(71\) 13.6154 1.61585 0.807924 0.589286i \(-0.200591\pi\)
0.807924 + 0.589286i \(0.200591\pi\)
\(72\) 4.63646 0.546412
\(73\) 0.439855 0.0514811 0.0257406 0.999669i \(-0.491806\pi\)
0.0257406 + 0.999669i \(0.491806\pi\)
\(74\) −8.07935 −0.939205
\(75\) 3.12695 0.361069
\(76\) −1.44748 −0.166038
\(77\) 0 0
\(78\) −0.430166 −0.0487067
\(79\) −17.0107 −1.91385 −0.956924 0.290338i \(-0.906232\pi\)
−0.956924 + 0.290338i \(0.906232\pi\)
\(80\) 2.47620 0.276847
\(81\) −1.41260 −0.156955
\(82\) −4.66518 −0.515183
\(83\) −6.10346 −0.669942 −0.334971 0.942229i \(-0.608726\pi\)
−0.334971 + 0.942229i \(0.608726\pi\)
\(84\) 0 0
\(85\) −12.9743 −1.40726
\(86\) −3.63646 −0.392130
\(87\) 2.76342 0.296269
\(88\) 4.76342 0.507782
\(89\) −10.1920 −1.08035 −0.540175 0.841553i \(-0.681642\pi\)
−0.540175 + 0.841553i \(0.681642\pi\)
\(90\) 11.4808 1.21018
\(91\) 0 0
\(92\) 0.552516 0.0576037
\(93\) −20.5889 −2.13497
\(94\) −4.55554 −0.469868
\(95\) −3.58426 −0.367737
\(96\) 2.76342 0.282040
\(97\) 4.66518 0.473677 0.236838 0.971549i \(-0.423889\pi\)
0.236838 + 0.971549i \(0.423889\pi\)
\(98\) 0 0
\(99\) 22.0854 2.21967
\(100\) 1.13155 0.113155
\(101\) 8.39988 0.835819 0.417910 0.908489i \(-0.362763\pi\)
0.417910 + 0.908489i \(0.362763\pi\)
\(102\) −14.4792 −1.43366
\(103\) 12.7331 1.25463 0.627316 0.778765i \(-0.284153\pi\)
0.627316 + 0.778765i \(0.284153\pi\)
\(104\) −0.155665 −0.0152642
\(105\) 0 0
\(106\) −7.16329 −0.695761
\(107\) 1.36814 0.132263 0.0661314 0.997811i \(-0.478934\pi\)
0.0661314 + 0.997811i \(0.478934\pi\)
\(108\) 4.52223 0.435152
\(109\) 8.53143 0.817163 0.408582 0.912722i \(-0.366024\pi\)
0.408582 + 0.912722i \(0.366024\pi\)
\(110\) 11.7952 1.12462
\(111\) −22.3266 −2.11915
\(112\) 0 0
\(113\) −14.0061 −1.31758 −0.658789 0.752327i \(-0.728931\pi\)
−0.658789 + 0.752327i \(0.728931\pi\)
\(114\) −4.00000 −0.374634
\(115\) 1.36814 0.127580
\(116\) 1.00000 0.0928477
\(117\) −0.721733 −0.0667243
\(118\) −6.05523 −0.557430
\(119\) 0 0
\(120\) 6.84276 0.624656
\(121\) 11.6901 1.06274
\(122\) 9.12695 0.826315
\(123\) −12.8918 −1.16242
\(124\) −7.45051 −0.669076
\(125\) −9.57904 −0.856775
\(126\) 0 0
\(127\) 18.4792 1.63977 0.819883 0.572531i \(-0.194038\pi\)
0.819883 + 0.572531i \(0.194038\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.0491 −0.884770
\(130\) −0.385456 −0.0338068
\(131\) 14.2472 1.24479 0.622394 0.782704i \(-0.286160\pi\)
0.622394 + 0.782704i \(0.286160\pi\)
\(132\) 13.1633 1.14572
\(133\) 0 0
\(134\) 14.7906 1.27771
\(135\) 11.1979 0.963764
\(136\) −5.23961 −0.449293
\(137\) 11.8382 1.01140 0.505701 0.862709i \(-0.331234\pi\)
0.505701 + 0.862709i \(0.331234\pi\)
\(138\) 1.52683 0.129972
\(139\) −3.31290 −0.280997 −0.140498 0.990081i \(-0.544870\pi\)
−0.140498 + 0.990081i \(0.544870\pi\)
\(140\) 0 0
\(141\) −12.5889 −1.06017
\(142\) 13.6154 1.14258
\(143\) −0.741495 −0.0620069
\(144\) 4.63646 0.386372
\(145\) 2.47620 0.205637
\(146\) 0.439855 0.0364026
\(147\) 0 0
\(148\) −8.07935 −0.664118
\(149\) 11.7377 0.961592 0.480796 0.876832i \(-0.340348\pi\)
0.480796 + 0.876832i \(0.340348\pi\)
\(150\) 3.12695 0.255315
\(151\) 21.5110 1.75054 0.875269 0.483637i \(-0.160684\pi\)
0.875269 + 0.483637i \(0.160684\pi\)
\(152\) −1.44748 −0.117406
\(153\) −24.2933 −1.96399
\(154\) 0 0
\(155\) −18.4489 −1.48185
\(156\) −0.430166 −0.0344408
\(157\) 22.2380 1.77479 0.887394 0.461011i \(-0.152513\pi\)
0.887394 + 0.461011i \(0.152513\pi\)
\(158\) −17.0107 −1.35330
\(159\) −19.7952 −1.56986
\(160\) 2.47620 0.195761
\(161\) 0 0
\(162\) −1.41260 −0.110984
\(163\) −18.4489 −1.44503 −0.722516 0.691354i \(-0.757015\pi\)
−0.722516 + 0.691354i \(0.757015\pi\)
\(164\) −4.66518 −0.364289
\(165\) 32.5949 2.53751
\(166\) −6.10346 −0.473720
\(167\) 21.5995 1.67142 0.835710 0.549170i \(-0.185056\pi\)
0.835710 + 0.549170i \(0.185056\pi\)
\(168\) 0 0
\(169\) −12.9758 −0.998136
\(170\) −12.9743 −0.995085
\(171\) −6.71121 −0.513219
\(172\) −3.63646 −0.277278
\(173\) 1.04980 0.0798146 0.0399073 0.999203i \(-0.487294\pi\)
0.0399073 + 0.999203i \(0.487294\pi\)
\(174\) 2.76342 0.209494
\(175\) 0 0
\(176\) 4.76342 0.359056
\(177\) −16.7331 −1.25774
\(178\) −10.1920 −0.763923
\(179\) 20.1587 1.50673 0.753366 0.657602i \(-0.228429\pi\)
0.753366 + 0.657602i \(0.228429\pi\)
\(180\) 11.4808 0.855728
\(181\) −0.155665 −0.0115705 −0.00578523 0.999983i \(-0.501842\pi\)
−0.00578523 + 0.999983i \(0.501842\pi\)
\(182\) 0 0
\(183\) 25.2216 1.86443
\(184\) 0.552516 0.0407320
\(185\) −20.0061 −1.47087
\(186\) −20.5889 −1.50965
\(187\) −24.9585 −1.82514
\(188\) −4.55554 −0.332247
\(189\) 0 0
\(190\) −3.58426 −0.260029
\(191\) −8.73313 −0.631907 −0.315953 0.948775i \(-0.602324\pi\)
−0.315953 + 0.948775i \(0.602324\pi\)
\(192\) 2.76342 0.199432
\(193\) −0.742332 −0.0534342 −0.0267171 0.999643i \(-0.508505\pi\)
−0.0267171 + 0.999643i \(0.508505\pi\)
\(194\) 4.66518 0.334940
\(195\) −1.06518 −0.0762788
\(196\) 0 0
\(197\) −10.2692 −0.731648 −0.365824 0.930684i \(-0.619213\pi\)
−0.365824 + 0.930684i \(0.619213\pi\)
\(198\) 22.0854 1.56954
\(199\) −0.574436 −0.0407207 −0.0203603 0.999793i \(-0.506481\pi\)
−0.0203603 + 0.999793i \(0.506481\pi\)
\(200\) 1.13155 0.0800129
\(201\) 40.8724 2.88292
\(202\) 8.39988 0.591013
\(203\) 0 0
\(204\) −14.4792 −1.01375
\(205\) −11.5519 −0.806820
\(206\) 12.7331 0.887159
\(207\) 2.56172 0.178052
\(208\) −0.155665 −0.0107934
\(209\) −6.89497 −0.476935
\(210\) 0 0
\(211\) −13.4264 −0.924312 −0.462156 0.886799i \(-0.652924\pi\)
−0.462156 + 0.886799i \(0.652924\pi\)
\(212\) −7.16329 −0.491977
\(213\) 37.6249 2.57802
\(214\) 1.36814 0.0935240
\(215\) −9.00460 −0.614109
\(216\) 4.52223 0.307699
\(217\) 0 0
\(218\) 8.53143 0.577821
\(219\) 1.21550 0.0821360
\(220\) 11.7952 0.795229
\(221\) 0.815622 0.0548647
\(222\) −22.3266 −1.49846
\(223\) −5.67947 −0.380325 −0.190163 0.981753i \(-0.560902\pi\)
−0.190163 + 0.981753i \(0.560902\pi\)
\(224\) 0 0
\(225\) 5.24641 0.349760
\(226\) −14.0061 −0.931669
\(227\) 20.9124 1.38801 0.694003 0.719972i \(-0.255845\pi\)
0.694003 + 0.719972i \(0.255845\pi\)
\(228\) −4.00000 −0.264906
\(229\) −22.6598 −1.49740 −0.748702 0.662907i \(-0.769323\pi\)
−0.748702 + 0.662907i \(0.769323\pi\)
\(230\) 1.36814 0.0902124
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) 18.9002 1.23819 0.619096 0.785315i \(-0.287499\pi\)
0.619096 + 0.785315i \(0.287499\pi\)
\(234\) −0.721733 −0.0471812
\(235\) −11.2804 −0.735854
\(236\) −6.05523 −0.394162
\(237\) −47.0075 −3.05347
\(238\) 0 0
\(239\) −24.4945 −1.58442 −0.792208 0.610251i \(-0.791069\pi\)
−0.792208 + 0.610251i \(0.791069\pi\)
\(240\) 6.84276 0.441698
\(241\) 29.2268 1.88266 0.941331 0.337485i \(-0.109576\pi\)
0.941331 + 0.337485i \(0.109576\pi\)
\(242\) 11.6901 0.751470
\(243\) −17.4703 −1.12072
\(244\) 9.12695 0.584293
\(245\) 0 0
\(246\) −12.8918 −0.821952
\(247\) 0.225322 0.0143369
\(248\) −7.45051 −0.473108
\(249\) −16.8664 −1.06886
\(250\) −9.57904 −0.605832
\(251\) −6.45209 −0.407252 −0.203626 0.979049i \(-0.565273\pi\)
−0.203626 + 0.979049i \(0.565273\pi\)
\(252\) 0 0
\(253\) 2.63186 0.165464
\(254\) 18.4792 1.15949
\(255\) −35.8534 −2.24523
\(256\) 1.00000 0.0625000
\(257\) −21.2646 −1.32645 −0.663223 0.748421i \(-0.730812\pi\)
−0.663223 + 0.748421i \(0.730812\pi\)
\(258\) −10.0491 −0.625627
\(259\) 0 0
\(260\) −0.385456 −0.0239050
\(261\) 4.63646 0.286990
\(262\) 14.2472 0.880198
\(263\) −29.5473 −1.82197 −0.910983 0.412445i \(-0.864675\pi\)
−0.910983 + 0.412445i \(0.864675\pi\)
\(264\) 13.1633 0.810145
\(265\) −17.7377 −1.08962
\(266\) 0 0
\(267\) −28.1647 −1.72365
\(268\) 14.7906 0.903477
\(269\) 7.45354 0.454450 0.227225 0.973842i \(-0.427035\pi\)
0.227225 + 0.973842i \(0.427035\pi\)
\(270\) 11.1979 0.681484
\(271\) −10.1490 −0.616508 −0.308254 0.951304i \(-0.599745\pi\)
−0.308254 + 0.951304i \(0.599745\pi\)
\(272\) −5.23961 −0.317698
\(273\) 0 0
\(274\) 11.8382 0.715169
\(275\) 5.39006 0.325033
\(276\) 1.52683 0.0919044
\(277\) 4.36437 0.262230 0.131115 0.991367i \(-0.458144\pi\)
0.131115 + 0.991367i \(0.458144\pi\)
\(278\) −3.31290 −0.198695
\(279\) −34.5440 −2.06810
\(280\) 0 0
\(281\) 18.0802 1.07857 0.539287 0.842122i \(-0.318694\pi\)
0.539287 + 0.842122i \(0.318694\pi\)
\(282\) −12.5889 −0.749656
\(283\) 31.3250 1.86208 0.931039 0.364919i \(-0.118903\pi\)
0.931039 + 0.364919i \(0.118903\pi\)
\(284\) 13.6154 0.807924
\(285\) −9.90479 −0.586709
\(286\) −0.741495 −0.0438455
\(287\) 0 0
\(288\) 4.63646 0.273206
\(289\) 10.4535 0.614914
\(290\) 2.47620 0.145407
\(291\) 12.8918 0.755732
\(292\) 0.439855 0.0257406
\(293\) 6.80642 0.397635 0.198818 0.980037i \(-0.436290\pi\)
0.198818 + 0.980037i \(0.436290\pi\)
\(294\) 0 0
\(295\) −14.9940 −0.872982
\(296\) −8.07935 −0.469602
\(297\) 21.5413 1.24995
\(298\) 11.7377 0.679948
\(299\) −0.0860072 −0.00497392
\(300\) 3.12695 0.180535
\(301\) 0 0
\(302\) 21.5110 1.23782
\(303\) 23.2124 1.33351
\(304\) −1.44748 −0.0830189
\(305\) 22.6001 1.29408
\(306\) −24.2933 −1.38875
\(307\) 1.39528 0.0796327 0.0398163 0.999207i \(-0.487323\pi\)
0.0398163 + 0.999207i \(0.487323\pi\)
\(308\) 0 0
\(309\) 35.1869 2.00171
\(310\) −18.4489 −1.04783
\(311\) −14.8178 −0.840241 −0.420121 0.907468i \(-0.638012\pi\)
−0.420121 + 0.907468i \(0.638012\pi\)
\(312\) −0.430166 −0.0243534
\(313\) 1.41720 0.0801047 0.0400524 0.999198i \(-0.487248\pi\)
0.0400524 + 0.999198i \(0.487248\pi\)
\(314\) 22.2380 1.25497
\(315\) 0 0
\(316\) −17.0107 −0.956924
\(317\) 12.7045 0.713558 0.356779 0.934189i \(-0.383875\pi\)
0.356779 + 0.934189i \(0.383875\pi\)
\(318\) −19.7952 −1.11006
\(319\) 4.76342 0.266700
\(320\) 2.47620 0.138424
\(321\) 3.78073 0.211020
\(322\) 0 0
\(323\) 7.58426 0.421999
\(324\) −1.41260 −0.0784775
\(325\) −0.176143 −0.00977064
\(326\) −18.4489 −1.02179
\(327\) 23.5759 1.30375
\(328\) −4.66518 −0.257591
\(329\) 0 0
\(330\) 32.5949 1.79429
\(331\) −20.4362 −1.12328 −0.561638 0.827383i \(-0.689829\pi\)
−0.561638 + 0.827383i \(0.689829\pi\)
\(332\) −6.10346 −0.334971
\(333\) −37.4596 −2.05277
\(334\) 21.5995 1.18187
\(335\) 36.6243 2.00100
\(336\) 0 0
\(337\) −9.35893 −0.509814 −0.254907 0.966966i \(-0.582045\pi\)
−0.254907 + 0.966966i \(0.582045\pi\)
\(338\) −12.9758 −0.705789
\(339\) −38.7045 −2.10214
\(340\) −12.9743 −0.703631
\(341\) −35.4899 −1.92188
\(342\) −6.71121 −0.362901
\(343\) 0 0
\(344\) −3.63646 −0.196065
\(345\) 3.78073 0.203548
\(346\) 1.04980 0.0564374
\(347\) −16.3113 −0.875638 −0.437819 0.899063i \(-0.644249\pi\)
−0.437819 + 0.899063i \(0.644249\pi\)
\(348\) 2.76342 0.148135
\(349\) 26.2630 1.40583 0.702913 0.711276i \(-0.251882\pi\)
0.702913 + 0.711276i \(0.251882\pi\)
\(350\) 0 0
\(351\) −0.703951 −0.0375741
\(352\) 4.76342 0.253891
\(353\) −8.89497 −0.473431 −0.236716 0.971579i \(-0.576071\pi\)
−0.236716 + 0.971579i \(0.576071\pi\)
\(354\) −16.7331 −0.889356
\(355\) 33.7144 1.78937
\(356\) −10.1920 −0.540175
\(357\) 0 0
\(358\) 20.1587 1.06542
\(359\) 1.88431 0.0994501 0.0497251 0.998763i \(-0.484165\pi\)
0.0497251 + 0.998763i \(0.484165\pi\)
\(360\) 11.4808 0.605091
\(361\) −16.9048 −0.889726
\(362\) −0.155665 −0.00818155
\(363\) 32.3047 1.69556
\(364\) 0 0
\(365\) 1.08917 0.0570096
\(366\) 25.2216 1.31835
\(367\) 19.0870 0.996332 0.498166 0.867082i \(-0.334007\pi\)
0.498166 + 0.867082i \(0.334007\pi\)
\(368\) 0.552516 0.0288019
\(369\) −21.6299 −1.12601
\(370\) −20.0061 −1.04007
\(371\) 0 0
\(372\) −20.5889 −1.06748
\(373\) 14.4423 0.747793 0.373896 0.927470i \(-0.378022\pi\)
0.373896 + 0.927470i \(0.378022\pi\)
\(374\) −24.9585 −1.29057
\(375\) −26.4709 −1.36695
\(376\) −4.55554 −0.234934
\(377\) −0.155665 −0.00801714
\(378\) 0 0
\(379\) 13.4847 0.692661 0.346330 0.938113i \(-0.387428\pi\)
0.346330 + 0.938113i \(0.387428\pi\)
\(380\) −3.58426 −0.183868
\(381\) 51.0658 2.61618
\(382\) −8.73313 −0.446826
\(383\) 3.57820 0.182838 0.0914188 0.995813i \(-0.470860\pi\)
0.0914188 + 0.995813i \(0.470860\pi\)
\(384\) 2.76342 0.141020
\(385\) 0 0
\(386\) −0.742332 −0.0377837
\(387\) −16.8603 −0.857059
\(388\) 4.66518 0.236838
\(389\) −6.55252 −0.332226 −0.166113 0.986107i \(-0.553122\pi\)
−0.166113 + 0.986107i \(0.553122\pi\)
\(390\) −1.06518 −0.0539373
\(391\) −2.89497 −0.146405
\(392\) 0 0
\(393\) 39.3710 1.98601
\(394\) −10.2692 −0.517353
\(395\) −42.1217 −2.11937
\(396\) 22.0854 1.10983
\(397\) 31.0751 1.55961 0.779807 0.626020i \(-0.215317\pi\)
0.779807 + 0.626020i \(0.215317\pi\)
\(398\) −0.574436 −0.0287939
\(399\) 0 0
\(400\) 1.13155 0.0565777
\(401\) −25.8678 −1.29178 −0.645889 0.763431i \(-0.723513\pi\)
−0.645889 + 0.763431i \(0.723513\pi\)
\(402\) 40.8724 2.03853
\(403\) 1.15978 0.0577728
\(404\) 8.39988 0.417910
\(405\) −3.49787 −0.173810
\(406\) 0 0
\(407\) −38.4853 −1.90764
\(408\) −14.4792 −0.716828
\(409\) 1.61152 0.0796843 0.0398422 0.999206i \(-0.487314\pi\)
0.0398422 + 0.999206i \(0.487314\pi\)
\(410\) −11.5519 −0.570508
\(411\) 32.7138 1.61365
\(412\) 12.7331 0.627316
\(413\) 0 0
\(414\) 2.56172 0.125902
\(415\) −15.1134 −0.741886
\(416\) −0.155665 −0.00763209
\(417\) −9.15493 −0.448319
\(418\) −6.89497 −0.337244
\(419\) 3.62423 0.177055 0.0885277 0.996074i \(-0.471784\pi\)
0.0885277 + 0.996074i \(0.471784\pi\)
\(420\) 0 0
\(421\) 10.0280 0.488734 0.244367 0.969683i \(-0.421420\pi\)
0.244367 + 0.969683i \(0.421420\pi\)
\(422\) −13.4264 −0.653587
\(423\) −21.1216 −1.02697
\(424\) −7.16329 −0.347880
\(425\) −5.92890 −0.287594
\(426\) 37.6249 1.82293
\(427\) 0 0
\(428\) 1.36814 0.0661314
\(429\) −2.04906 −0.0989295
\(430\) −9.00460 −0.434240
\(431\) 3.57987 0.172436 0.0862182 0.996276i \(-0.472522\pi\)
0.0862182 + 0.996276i \(0.472522\pi\)
\(432\) 4.52223 0.217576
\(433\) 12.5125 0.601314 0.300657 0.953732i \(-0.402794\pi\)
0.300657 + 0.953732i \(0.402794\pi\)
\(434\) 0 0
\(435\) 6.84276 0.328085
\(436\) 8.53143 0.408582
\(437\) −0.799758 −0.0382576
\(438\) 1.21550 0.0580789
\(439\) −22.7906 −1.08773 −0.543867 0.839171i \(-0.683040\pi\)
−0.543867 + 0.839171i \(0.683040\pi\)
\(440\) 11.7952 0.562312
\(441\) 0 0
\(442\) 0.815622 0.0387952
\(443\) 25.6636 1.21931 0.609657 0.792665i \(-0.291307\pi\)
0.609657 + 0.792665i \(0.291307\pi\)
\(444\) −22.3266 −1.05957
\(445\) −25.2374 −1.19637
\(446\) −5.67947 −0.268931
\(447\) 32.4362 1.53418
\(448\) 0 0
\(449\) −6.07269 −0.286588 −0.143294 0.989680i \(-0.545769\pi\)
−0.143294 + 0.989680i \(0.545769\pi\)
\(450\) 5.24641 0.247318
\(451\) −22.2222 −1.04640
\(452\) −14.0061 −0.658789
\(453\) 59.4437 2.79291
\(454\) 20.9124 0.981468
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 12.3485 0.577639 0.288819 0.957384i \(-0.406737\pi\)
0.288819 + 0.957384i \(0.406737\pi\)
\(458\) −22.6598 −1.05882
\(459\) −23.6947 −1.10598
\(460\) 1.36814 0.0637898
\(461\) 32.6024 1.51845 0.759223 0.650831i \(-0.225579\pi\)
0.759223 + 0.650831i \(0.225579\pi\)
\(462\) 0 0
\(463\) 9.73023 0.452202 0.226101 0.974104i \(-0.427402\pi\)
0.226101 + 0.974104i \(0.427402\pi\)
\(464\) 1.00000 0.0464238
\(465\) −50.9821 −2.36424
\(466\) 18.9002 0.875534
\(467\) 33.0234 1.52814 0.764070 0.645134i \(-0.223198\pi\)
0.764070 + 0.645134i \(0.223198\pi\)
\(468\) −0.721733 −0.0333621
\(469\) 0 0
\(470\) −11.2804 −0.520327
\(471\) 61.4529 2.83160
\(472\) −6.05523 −0.278715
\(473\) −17.3220 −0.796466
\(474\) −47.0075 −2.15913
\(475\) −1.63791 −0.0751523
\(476\) 0 0
\(477\) −33.2124 −1.52069
\(478\) −24.4945 −1.12035
\(479\) −5.96752 −0.272663 −0.136332 0.990663i \(-0.543531\pi\)
−0.136332 + 0.990663i \(0.543531\pi\)
\(480\) 6.84276 0.312328
\(481\) 1.25767 0.0573447
\(482\) 29.2268 1.33124
\(483\) 0 0
\(484\) 11.6901 0.531369
\(485\) 11.5519 0.524545
\(486\) −17.4703 −0.792468
\(487\) 22.4126 1.01561 0.507806 0.861472i \(-0.330457\pi\)
0.507806 + 0.861472i \(0.330457\pi\)
\(488\) 9.12695 0.413158
\(489\) −50.9821 −2.30549
\(490\) 0 0
\(491\) 12.5949 0.568401 0.284200 0.958765i \(-0.408272\pi\)
0.284200 + 0.958765i \(0.408272\pi\)
\(492\) −12.8918 −0.581208
\(493\) −5.23961 −0.235980
\(494\) 0.225322 0.0101377
\(495\) 54.6878 2.45803
\(496\) −7.45051 −0.334538
\(497\) 0 0
\(498\) −16.8664 −0.755801
\(499\) −39.7490 −1.77941 −0.889705 0.456536i \(-0.849090\pi\)
−0.889705 + 0.456536i \(0.849090\pi\)
\(500\) −9.57904 −0.428388
\(501\) 59.6884 2.66668
\(502\) −6.45209 −0.287971
\(503\) −0.561599 −0.0250405 −0.0125202 0.999922i \(-0.503985\pi\)
−0.0125202 + 0.999922i \(0.503985\pi\)
\(504\) 0 0
\(505\) 20.7998 0.925577
\(506\) 2.63186 0.117001
\(507\) −35.8574 −1.59248
\(508\) 18.4792 0.819883
\(509\) −39.8126 −1.76466 −0.882331 0.470629i \(-0.844027\pi\)
−0.882331 + 0.470629i \(0.844027\pi\)
\(510\) −35.8534 −1.58762
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −6.54585 −0.289007
\(514\) −21.2646 −0.937940
\(515\) 31.5297 1.38937
\(516\) −10.0491 −0.442385
\(517\) −21.6999 −0.954363
\(518\) 0 0
\(519\) 2.90102 0.127341
\(520\) −0.385456 −0.0169034
\(521\) −12.2017 −0.534566 −0.267283 0.963618i \(-0.586126\pi\)
−0.267283 + 0.963618i \(0.586126\pi\)
\(522\) 4.63646 0.202933
\(523\) −14.8305 −0.648494 −0.324247 0.945972i \(-0.605111\pi\)
−0.324247 + 0.945972i \(0.605111\pi\)
\(524\) 14.2472 0.622394
\(525\) 0 0
\(526\) −29.5473 −1.28832
\(527\) 39.0378 1.70051
\(528\) 13.1633 0.572859
\(529\) −22.6947 −0.986727
\(530\) −17.7377 −0.770478
\(531\) −28.0749 −1.21835
\(532\) 0 0
\(533\) 0.726203 0.0314553
\(534\) −28.1647 −1.21881
\(535\) 3.38778 0.146466
\(536\) 14.7906 0.638855
\(537\) 55.7068 2.40393
\(538\) 7.45354 0.321345
\(539\) 0 0
\(540\) 11.1979 0.481882
\(541\) −0.588022 −0.0252810 −0.0126405 0.999920i \(-0.504024\pi\)
−0.0126405 + 0.999920i \(0.504024\pi\)
\(542\) −10.1490 −0.435937
\(543\) −0.430166 −0.0184602
\(544\) −5.23961 −0.224647
\(545\) 21.1255 0.904917
\(546\) 0 0
\(547\) 19.0159 0.813060 0.406530 0.913637i \(-0.366739\pi\)
0.406530 + 0.913637i \(0.366739\pi\)
\(548\) 11.8382 0.505701
\(549\) 42.3168 1.80604
\(550\) 5.39006 0.229833
\(551\) −1.44748 −0.0616649
\(552\) 1.52683 0.0649862
\(553\) 0 0
\(554\) 4.36437 0.185424
\(555\) −55.2850 −2.34672
\(556\) −3.31290 −0.140498
\(557\) 0.253903 0.0107582 0.00537911 0.999986i \(-0.498288\pi\)
0.00537911 + 0.999986i \(0.498288\pi\)
\(558\) −34.5440 −1.46237
\(559\) 0.566069 0.0239422
\(560\) 0 0
\(561\) −68.9706 −2.91194
\(562\) 18.0802 0.762666
\(563\) −29.1094 −1.22681 −0.613407 0.789767i \(-0.710201\pi\)
−0.613407 + 0.789767i \(0.710201\pi\)
\(564\) −12.5889 −0.530087
\(565\) −34.6818 −1.45907
\(566\) 31.3250 1.31669
\(567\) 0 0
\(568\) 13.6154 0.571289
\(569\) −37.7068 −1.58075 −0.790376 0.612621i \(-0.790115\pi\)
−0.790376 + 0.612621i \(0.790115\pi\)
\(570\) −9.90479 −0.414866
\(571\) 32.3418 1.35346 0.676732 0.736229i \(-0.263396\pi\)
0.676732 + 0.736229i \(0.263396\pi\)
\(572\) −0.741495 −0.0310035
\(573\) −24.1333 −1.00818
\(574\) 0 0
\(575\) 0.625201 0.0260727
\(576\) 4.63646 0.193186
\(577\) 10.1438 0.422291 0.211146 0.977455i \(-0.432281\pi\)
0.211146 + 0.977455i \(0.432281\pi\)
\(578\) 10.4535 0.434810
\(579\) −2.05137 −0.0852521
\(580\) 2.47620 0.102818
\(581\) 0 0
\(582\) 12.8918 0.534383
\(583\) −34.1217 −1.41318
\(584\) 0.439855 0.0182013
\(585\) −1.78715 −0.0738897
\(586\) 6.80642 0.281171
\(587\) −26.4453 −1.09151 −0.545757 0.837943i \(-0.683758\pi\)
−0.545757 + 0.837943i \(0.683758\pi\)
\(588\) 0 0
\(589\) 10.7845 0.444368
\(590\) −14.9940 −0.617291
\(591\) −28.3780 −1.16731
\(592\) −8.07935 −0.332059
\(593\) 5.83900 0.239779 0.119889 0.992787i \(-0.461746\pi\)
0.119889 + 0.992787i \(0.461746\pi\)
\(594\) 21.5413 0.883848
\(595\) 0 0
\(596\) 11.7377 0.480796
\(597\) −1.58740 −0.0649681
\(598\) −0.0860072 −0.00351709
\(599\) −10.1157 −0.413316 −0.206658 0.978413i \(-0.566259\pi\)
−0.206658 + 0.978413i \(0.566259\pi\)
\(600\) 3.12695 0.127657
\(601\) −34.5954 −1.41118 −0.705588 0.708622i \(-0.749317\pi\)
−0.705588 + 0.708622i \(0.749317\pi\)
\(602\) 0 0
\(603\) 68.5759 2.79263
\(604\) 21.5110 0.875269
\(605\) 28.9471 1.17687
\(606\) 23.2124 0.942937
\(607\) −10.8339 −0.439735 −0.219867 0.975530i \(-0.570562\pi\)
−0.219867 + 0.975530i \(0.570562\pi\)
\(608\) −1.44748 −0.0587032
\(609\) 0 0
\(610\) 22.6001 0.915053
\(611\) 0.709137 0.0286886
\(612\) −24.2933 −0.981997
\(613\) 6.67487 0.269595 0.134798 0.990873i \(-0.456962\pi\)
0.134798 + 0.990873i \(0.456962\pi\)
\(614\) 1.39528 0.0563088
\(615\) −31.9227 −1.28725
\(616\) 0 0
\(617\) 11.7039 0.471182 0.235591 0.971852i \(-0.424297\pi\)
0.235591 + 0.971852i \(0.424297\pi\)
\(618\) 35.1869 1.41543
\(619\) −28.2525 −1.13556 −0.567781 0.823180i \(-0.692198\pi\)
−0.567781 + 0.823180i \(0.692198\pi\)
\(620\) −18.4489 −0.740927
\(621\) 2.49860 0.100265
\(622\) −14.8178 −0.594140
\(623\) 0 0
\(624\) −0.430166 −0.0172204
\(625\) −29.3774 −1.17509
\(626\) 1.41720 0.0566426
\(627\) −19.0537 −0.760930
\(628\) 22.2380 0.887394
\(629\) 42.3326 1.68791
\(630\) 0 0
\(631\) −45.3197 −1.80415 −0.902074 0.431582i \(-0.857956\pi\)
−0.902074 + 0.431582i \(0.857956\pi\)
\(632\) −17.0107 −0.676648
\(633\) −37.1027 −1.47470
\(634\) 12.7045 0.504562
\(635\) 45.7582 1.81586
\(636\) −19.7952 −0.784929
\(637\) 0 0
\(638\) 4.76342 0.188585
\(639\) 63.1272 2.49727
\(640\) 2.47620 0.0978803
\(641\) −9.38778 −0.370795 −0.185398 0.982664i \(-0.559357\pi\)
−0.185398 + 0.982664i \(0.559357\pi\)
\(642\) 3.78073 0.149214
\(643\) −17.5246 −0.691104 −0.345552 0.938400i \(-0.612308\pi\)
−0.345552 + 0.938400i \(0.612308\pi\)
\(644\) 0 0
\(645\) −24.8835 −0.979785
\(646\) 7.58426 0.298399
\(647\) 32.0603 1.26042 0.630211 0.776424i \(-0.282968\pi\)
0.630211 + 0.776424i \(0.282968\pi\)
\(648\) −1.41260 −0.0554920
\(649\) −28.8436 −1.13221
\(650\) −0.176143 −0.00690889
\(651\) 0 0
\(652\) −18.4489 −0.722516
\(653\) −17.5903 −0.688362 −0.344181 0.938903i \(-0.611843\pi\)
−0.344181 + 0.938903i \(0.611843\pi\)
\(654\) 23.5759 0.921890
\(655\) 35.2790 1.37846
\(656\) −4.66518 −0.182145
\(657\) 2.03937 0.0795634
\(658\) 0 0
\(659\) 22.2902 0.868305 0.434152 0.900839i \(-0.357048\pi\)
0.434152 + 0.900839i \(0.357048\pi\)
\(660\) 32.5949 1.26876
\(661\) −24.3288 −0.946280 −0.473140 0.880987i \(-0.656879\pi\)
−0.473140 + 0.880987i \(0.656879\pi\)
\(662\) −20.4362 −0.794276
\(663\) 2.25390 0.0875343
\(664\) −6.10346 −0.236860
\(665\) 0 0
\(666\) −37.4596 −1.45153
\(667\) 0.552516 0.0213935
\(668\) 21.5995 0.835710
\(669\) −15.6947 −0.606793
\(670\) 36.6243 1.41492
\(671\) 43.4755 1.67835
\(672\) 0 0
\(673\) −4.97515 −0.191778 −0.0958890 0.995392i \(-0.530569\pi\)
−0.0958890 + 0.995392i \(0.530569\pi\)
\(674\) −9.35893 −0.360493
\(675\) 5.11714 0.196959
\(676\) −12.9758 −0.499068
\(677\) −12.7839 −0.491325 −0.245662 0.969355i \(-0.579005\pi\)
−0.245662 + 0.969355i \(0.579005\pi\)
\(678\) −38.7045 −1.48644
\(679\) 0 0
\(680\) −12.9743 −0.497542
\(681\) 57.7897 2.21451
\(682\) −35.4899 −1.35898
\(683\) 25.1171 0.961081 0.480540 0.876973i \(-0.340441\pi\)
0.480540 + 0.876973i \(0.340441\pi\)
\(684\) −6.71121 −0.256609
\(685\) 29.3136 1.12002
\(686\) 0 0
\(687\) −62.6185 −2.38905
\(688\) −3.63646 −0.138639
\(689\) 1.11507 0.0424808
\(690\) 3.78073 0.143930
\(691\) −50.0191 −1.90282 −0.951409 0.307931i \(-0.900363\pi\)
−0.951409 + 0.307931i \(0.900363\pi\)
\(692\) 1.04980 0.0399073
\(693\) 0 0
\(694\) −16.3113 −0.619170
\(695\) −8.20340 −0.311173
\(696\) 2.76342 0.104747
\(697\) 24.4437 0.925872
\(698\) 26.2630 0.994069
\(699\) 52.2291 1.97548
\(700\) 0 0
\(701\) 20.3165 0.767345 0.383673 0.923469i \(-0.374659\pi\)
0.383673 + 0.923469i \(0.374659\pi\)
\(702\) −0.703951 −0.0265689
\(703\) 11.6947 0.441075
\(704\) 4.76342 0.179528
\(705\) −31.1725 −1.17402
\(706\) −8.89497 −0.334767
\(707\) 0 0
\(708\) −16.7331 −0.628869
\(709\) −26.0161 −0.977055 −0.488527 0.872549i \(-0.662466\pi\)
−0.488527 + 0.872549i \(0.662466\pi\)
\(710\) 33.7144 1.26528
\(711\) −78.8693 −2.95783
\(712\) −10.1920 −0.381962
\(713\) −4.11653 −0.154165
\(714\) 0 0
\(715\) −1.83609 −0.0686658
\(716\) 20.1587 0.753366
\(717\) −67.6884 −2.52787
\(718\) 1.88431 0.0703219
\(719\) 18.5519 0.691870 0.345935 0.938259i \(-0.387562\pi\)
0.345935 + 0.938259i \(0.387562\pi\)
\(720\) 11.4808 0.427864
\(721\) 0 0
\(722\) −16.9048 −0.629131
\(723\) 80.7657 3.00371
\(724\) −0.155665 −0.00578523
\(725\) 1.13155 0.0420248
\(726\) 32.3047 1.19894
\(727\) −1.02788 −0.0381218 −0.0190609 0.999818i \(-0.506068\pi\)
−0.0190609 + 0.999818i \(0.506068\pi\)
\(728\) 0 0
\(729\) −44.0398 −1.63110
\(730\) 1.08917 0.0403119
\(731\) 19.0537 0.704725
\(732\) 25.2216 0.932216
\(733\) 19.8632 0.733665 0.366833 0.930287i \(-0.380442\pi\)
0.366833 + 0.930287i \(0.380442\pi\)
\(734\) 19.0870 0.704513
\(735\) 0 0
\(736\) 0.552516 0.0203660
\(737\) 70.4536 2.59519
\(738\) −21.6299 −0.796208
\(739\) −18.9094 −0.695593 −0.347797 0.937570i \(-0.613070\pi\)
−0.347797 + 0.937570i \(0.613070\pi\)
\(740\) −20.0061 −0.735437
\(741\) 0.622658 0.0228739
\(742\) 0 0
\(743\) −16.5078 −0.605613 −0.302806 0.953052i \(-0.597924\pi\)
−0.302806 + 0.953052i \(0.597924\pi\)
\(744\) −20.5889 −0.754824
\(745\) 29.0649 1.06486
\(746\) 14.4423 0.528769
\(747\) −28.2985 −1.03539
\(748\) −24.9585 −0.912571
\(749\) 0 0
\(750\) −26.4709 −0.966579
\(751\) −29.1610 −1.06410 −0.532050 0.846713i \(-0.678578\pi\)
−0.532050 + 0.846713i \(0.678578\pi\)
\(752\) −4.55554 −0.166124
\(753\) −17.8298 −0.649754
\(754\) −0.155665 −0.00566897
\(755\) 53.2654 1.93853
\(756\) 0 0
\(757\) 36.9956 1.34463 0.672315 0.740266i \(-0.265300\pi\)
0.672315 + 0.740266i \(0.265300\pi\)
\(758\) 13.4847 0.489785
\(759\) 7.27293 0.263991
\(760\) −3.58426 −0.130015
\(761\) −47.1171 −1.70800 −0.853998 0.520277i \(-0.825829\pi\)
−0.853998 + 0.520277i \(0.825829\pi\)
\(762\) 51.0658 1.84992
\(763\) 0 0
\(764\) −8.73313 −0.315953
\(765\) −60.1549 −2.17491
\(766\) 3.57820 0.129286
\(767\) 0.942586 0.0340348
\(768\) 2.76342 0.0997162
\(769\) −6.52174 −0.235180 −0.117590 0.993062i \(-0.537517\pi\)
−0.117590 + 0.993062i \(0.537517\pi\)
\(770\) 0 0
\(771\) −58.7628 −2.11629
\(772\) −0.742332 −0.0267171
\(773\) 16.3332 0.587466 0.293733 0.955887i \(-0.405102\pi\)
0.293733 + 0.955887i \(0.405102\pi\)
\(774\) −16.8603 −0.606032
\(775\) −8.43065 −0.302838
\(776\) 4.66518 0.167470
\(777\) 0 0
\(778\) −6.55252 −0.234919
\(779\) 6.75277 0.241943
\(780\) −1.06518 −0.0381394
\(781\) 64.8557 2.32072
\(782\) −2.89497 −0.103524
\(783\) 4.52223 0.161611
\(784\) 0 0
\(785\) 55.0658 1.96538
\(786\) 39.3710 1.40432
\(787\) −38.0673 −1.35695 −0.678477 0.734621i \(-0.737360\pi\)
−0.678477 + 0.734621i \(0.737360\pi\)
\(788\) −10.2692 −0.365824
\(789\) −81.6515 −2.90687
\(790\) −42.1217 −1.49862
\(791\) 0 0
\(792\) 22.0854 0.784771
\(793\) −1.42074 −0.0504521
\(794\) 31.0751 1.10281
\(795\) −49.0167 −1.73844
\(796\) −0.574436 −0.0203603
\(797\) −35.7665 −1.26692 −0.633458 0.773777i \(-0.718365\pi\)
−0.633458 + 0.773777i \(0.718365\pi\)
\(798\) 0 0
\(799\) 23.8693 0.844435
\(800\) 1.13155 0.0400064
\(801\) −47.2549 −1.66967
\(802\) −25.8678 −0.913425
\(803\) 2.09521 0.0739384
\(804\) 40.8724 1.44146
\(805\) 0 0
\(806\) 1.15978 0.0408515
\(807\) 20.5972 0.725057
\(808\) 8.39988 0.295507
\(809\) 7.33035 0.257722 0.128861 0.991663i \(-0.458868\pi\)
0.128861 + 0.991663i \(0.458868\pi\)
\(810\) −3.49787 −0.122902
\(811\) −25.1377 −0.882705 −0.441353 0.897334i \(-0.645501\pi\)
−0.441353 + 0.897334i \(0.645501\pi\)
\(812\) 0 0
\(813\) −28.0459 −0.983613
\(814\) −38.4853 −1.34891
\(815\) −45.6832 −1.60021
\(816\) −14.4792 −0.506874
\(817\) 5.26372 0.184154
\(818\) 1.61152 0.0563453
\(819\) 0 0
\(820\) −11.5519 −0.403410
\(821\) 12.5857 0.439244 0.219622 0.975585i \(-0.429518\pi\)
0.219622 + 0.975585i \(0.429518\pi\)
\(822\) 32.7138 1.14102
\(823\) 5.08662 0.177309 0.0886543 0.996062i \(-0.471743\pi\)
0.0886543 + 0.996062i \(0.471743\pi\)
\(824\) 12.7331 0.443580
\(825\) 14.8950 0.518576
\(826\) 0 0
\(827\) 33.9244 1.17967 0.589833 0.807525i \(-0.299193\pi\)
0.589833 + 0.807525i \(0.299193\pi\)
\(828\) 2.56172 0.0890259
\(829\) 36.0762 1.25298 0.626489 0.779430i \(-0.284491\pi\)
0.626489 + 0.779430i \(0.284491\pi\)
\(830\) −15.1134 −0.524593
\(831\) 12.0606 0.418377
\(832\) −0.155665 −0.00539670
\(833\) 0 0
\(834\) −9.15493 −0.317009
\(835\) 53.4847 1.85091
\(836\) −6.89497 −0.238467
\(837\) −33.6929 −1.16460
\(838\) 3.62423 0.125197
\(839\) −29.0983 −1.00458 −0.502292 0.864698i \(-0.667510\pi\)
−0.502292 + 0.864698i \(0.667510\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 10.0280 0.345587
\(843\) 49.9631 1.72082
\(844\) −13.4264 −0.462156
\(845\) −32.1306 −1.10532
\(846\) −21.1216 −0.726176
\(847\) 0 0
\(848\) −7.16329 −0.245989
\(849\) 86.5640 2.97087
\(850\) −5.92890 −0.203360
\(851\) −4.46397 −0.153023
\(852\) 37.6249 1.28901
\(853\) −10.9169 −0.373787 −0.186894 0.982380i \(-0.559842\pi\)
−0.186894 + 0.982380i \(0.559842\pi\)
\(854\) 0 0
\(855\) −16.6183 −0.568333
\(856\) 1.36814 0.0467620
\(857\) 31.7421 1.08429 0.542145 0.840285i \(-0.317612\pi\)
0.542145 + 0.840285i \(0.317612\pi\)
\(858\) −2.04906 −0.0699537
\(859\) 11.5632 0.394530 0.197265 0.980350i \(-0.436794\pi\)
0.197265 + 0.980350i \(0.436794\pi\)
\(860\) −9.00460 −0.307054
\(861\) 0 0
\(862\) 3.57987 0.121931
\(863\) −31.4596 −1.07090 −0.535449 0.844568i \(-0.679857\pi\)
−0.535449 + 0.844568i \(0.679857\pi\)
\(864\) 4.52223 0.153849
\(865\) 2.59950 0.0883858
\(866\) 12.5125 0.425193
\(867\) 28.8875 0.981070
\(868\) 0 0
\(869\) −81.0288 −2.74871
\(870\) 6.84276 0.231991
\(871\) −2.30237 −0.0780127
\(872\) 8.53143 0.288911
\(873\) 21.6299 0.732062
\(874\) −0.799758 −0.0270522
\(875\) 0 0
\(876\) 1.21550 0.0410680
\(877\) −9.49715 −0.320696 −0.160348 0.987061i \(-0.551262\pi\)
−0.160348 + 0.987061i \(0.551262\pi\)
\(878\) −22.7906 −0.769144
\(879\) 18.8090 0.634411
\(880\) 11.7952 0.397615
\(881\) 33.2641 1.12070 0.560348 0.828258i \(-0.310668\pi\)
0.560348 + 0.828258i \(0.310668\pi\)
\(882\) 0 0
\(883\) 28.5805 0.961810 0.480905 0.876773i \(-0.340308\pi\)
0.480905 + 0.876773i \(0.340308\pi\)
\(884\) 0.815622 0.0274323
\(885\) −41.4345 −1.39281
\(886\) 25.6636 0.862186
\(887\) 17.5397 0.588924 0.294462 0.955663i \(-0.404859\pi\)
0.294462 + 0.955663i \(0.404859\pi\)
\(888\) −22.3266 −0.749231
\(889\) 0 0
\(890\) −25.2374 −0.845960
\(891\) −6.72878 −0.225423
\(892\) −5.67947 −0.190163
\(893\) 6.59408 0.220662
\(894\) 32.4362 1.08483
\(895\) 49.9169 1.66854
\(896\) 0 0
\(897\) −0.237673 −0.00793569
\(898\) −6.07269 −0.202648
\(899\) −7.45051 −0.248488
\(900\) 5.24641 0.174880
\(901\) 37.5329 1.25040
\(902\) −22.2222 −0.739917
\(903\) 0 0
\(904\) −14.0061 −0.465834
\(905\) −0.385456 −0.0128130
\(906\) 59.4437 1.97489
\(907\) −13.1558 −0.436831 −0.218416 0.975856i \(-0.570089\pi\)
−0.218416 + 0.975856i \(0.570089\pi\)
\(908\) 20.9124 0.694003
\(909\) 38.9457 1.29175
\(910\) 0 0
\(911\) −26.5797 −0.880623 −0.440312 0.897845i \(-0.645132\pi\)
−0.440312 + 0.897845i \(0.645132\pi\)
\(912\) −4.00000 −0.132453
\(913\) −29.0733 −0.962186
\(914\) 12.3485 0.408452
\(915\) 62.4536 2.06465
\(916\) −22.6598 −0.748702
\(917\) 0 0
\(918\) −23.6947 −0.782043
\(919\) 16.2472 0.535947 0.267974 0.963426i \(-0.413646\pi\)
0.267974 + 0.963426i \(0.413646\pi\)
\(920\) 1.36814 0.0451062
\(921\) 3.85573 0.127051
\(922\) 32.6024 1.07370
\(923\) −2.11943 −0.0697620
\(924\) 0 0
\(925\) −9.14221 −0.300594
\(926\) 9.73023 0.319755
\(927\) 59.0367 1.93902
\(928\) 1.00000 0.0328266
\(929\) 21.6763 0.711177 0.355588 0.934643i \(-0.384280\pi\)
0.355588 + 0.934643i \(0.384280\pi\)
\(930\) −50.9821 −1.67177
\(931\) 0 0
\(932\) 18.9002 0.619096
\(933\) −40.9478 −1.34057
\(934\) 33.0234 1.08056
\(935\) −61.8020 −2.02114
\(936\) −0.721733 −0.0235906
\(937\) −40.3688 −1.31879 −0.659395 0.751797i \(-0.729187\pi\)
−0.659395 + 0.751797i \(0.729187\pi\)
\(938\) 0 0
\(939\) 3.91630 0.127804
\(940\) −11.2804 −0.367927
\(941\) 20.8571 0.679921 0.339960 0.940440i \(-0.389586\pi\)
0.339960 + 0.940440i \(0.389586\pi\)
\(942\) 61.4529 2.00224
\(943\) −2.57758 −0.0839377
\(944\) −6.05523 −0.197081
\(945\) 0 0
\(946\) −17.3220 −0.563186
\(947\) −15.2163 −0.494465 −0.247232 0.968956i \(-0.579521\pi\)
−0.247232 + 0.968956i \(0.579521\pi\)
\(948\) −47.0075 −1.52673
\(949\) −0.0684698 −0.00222262
\(950\) −1.63791 −0.0531407
\(951\) 35.1079 1.13845
\(952\) 0 0
\(953\) −2.12235 −0.0687497 −0.0343748 0.999409i \(-0.510944\pi\)
−0.0343748 + 0.999409i \(0.510944\pi\)
\(954\) −33.2124 −1.07529
\(955\) −21.6249 −0.699767
\(956\) −24.4945 −0.792208
\(957\) 13.1633 0.425509
\(958\) −5.96752 −0.192802
\(959\) 0 0
\(960\) 6.84276 0.220849
\(961\) 24.5101 0.790649
\(962\) 1.25767 0.0405489
\(963\) 6.34332 0.204411
\(964\) 29.2268 0.941331
\(965\) −1.83816 −0.0591725
\(966\) 0 0
\(967\) −55.4039 −1.78167 −0.890834 0.454328i \(-0.849879\pi\)
−0.890834 + 0.454328i \(0.849879\pi\)
\(968\) 11.6901 0.375735
\(969\) 20.9585 0.673282
\(970\) 11.5519 0.370909
\(971\) −21.2888 −0.683190 −0.341595 0.939847i \(-0.610967\pi\)
−0.341595 + 0.939847i \(0.610967\pi\)
\(972\) −17.4703 −0.560359
\(973\) 0 0
\(974\) 22.4126 0.718146
\(975\) −0.486756 −0.0155887
\(976\) 9.12695 0.292147
\(977\) −18.7651 −0.600348 −0.300174 0.953884i \(-0.597045\pi\)
−0.300174 + 0.953884i \(0.597045\pi\)
\(978\) −50.9821 −1.63023
\(979\) −48.5488 −1.55163
\(980\) 0 0
\(981\) 39.5557 1.26292
\(982\) 12.5949 0.401920
\(983\) 33.1176 1.05629 0.528144 0.849155i \(-0.322888\pi\)
0.528144 + 0.849155i \(0.322888\pi\)
\(984\) −12.8918 −0.410976
\(985\) −25.4285 −0.810218
\(986\) −5.23961 −0.166863
\(987\) 0 0
\(988\) 0.225322 0.00716845
\(989\) −2.00920 −0.0638890
\(990\) 54.6878 1.73809
\(991\) −14.3450 −0.455684 −0.227842 0.973698i \(-0.573167\pi\)
−0.227842 + 0.973698i \(0.573167\pi\)
\(992\) −7.45051 −0.236554
\(993\) −56.4738 −1.79214
\(994\) 0 0
\(995\) −1.42242 −0.0450936
\(996\) −16.8664 −0.534432
\(997\) −20.3260 −0.643730 −0.321865 0.946786i \(-0.604310\pi\)
−0.321865 + 0.946786i \(0.604310\pi\)
\(998\) −39.7490 −1.25823
\(999\) −36.5367 −1.15597
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2842.2.a.r.1.4 4
7.6 odd 2 406.2.a.g.1.1 4
21.20 even 2 3654.2.a.bg.1.3 4
28.27 even 2 3248.2.a.x.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
406.2.a.g.1.1 4 7.6 odd 2
2842.2.a.r.1.4 4 1.1 even 1 trivial
3248.2.a.x.1.4 4 28.27 even 2
3654.2.a.bg.1.3 4 21.20 even 2